{"title":"离散线性系统中schwarz矩阵的性质","authors":"C.D. Lee, S. Koyuncu","doi":"10.37418/amsj.12.7.1","DOIUrl":null,"url":null,"abstract":"In this paper, we investigate the properties of the Schwarz matrix, a specific type of matrix that appears in the stability analysis of discrete-time linear time-invariant systems. We derive a formula for the determinant of the Schwarz matrix and a formula for its permanent. We also provide conditions on the entries of the Schwarz matrix that ensure the system described by the state update equation $x_{k+1} = Bx_k$ is stable, as well as conditions that guarantee the eigenvalues of the Schwarz matrix are real. These findings provide insights into the stability properties of systems characterized by Schwarz matrices and offer new tools for the analysis of interconnected subsystems in a cascaded structure.","PeriodicalId":231117,"journal":{"name":"Advances in Mathematics: Scientific Journal","volume":"8 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"PROPERTIES OF SCHWARZ MATRICES IN DISCRETE-TIME LINEAR SYSTEMS\",\"authors\":\"C.D. Lee, S. Koyuncu\",\"doi\":\"10.37418/amsj.12.7.1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we investigate the properties of the Schwarz matrix, a specific type of matrix that appears in the stability analysis of discrete-time linear time-invariant systems. We derive a formula for the determinant of the Schwarz matrix and a formula for its permanent. We also provide conditions on the entries of the Schwarz matrix that ensure the system described by the state update equation $x_{k+1} = Bx_k$ is stable, as well as conditions that guarantee the eigenvalues of the Schwarz matrix are real. These findings provide insights into the stability properties of systems characterized by Schwarz matrices and offer new tools for the analysis of interconnected subsystems in a cascaded structure.\",\"PeriodicalId\":231117,\"journal\":{\"name\":\"Advances in Mathematics: Scientific Journal\",\"volume\":\"8 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-07-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Mathematics: Scientific Journal\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.37418/amsj.12.7.1\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Mathematics: Scientific Journal","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.37418/amsj.12.7.1","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
PROPERTIES OF SCHWARZ MATRICES IN DISCRETE-TIME LINEAR SYSTEMS
In this paper, we investigate the properties of the Schwarz matrix, a specific type of matrix that appears in the stability analysis of discrete-time linear time-invariant systems. We derive a formula for the determinant of the Schwarz matrix and a formula for its permanent. We also provide conditions on the entries of the Schwarz matrix that ensure the system described by the state update equation $x_{k+1} = Bx_k$ is stable, as well as conditions that guarantee the eigenvalues of the Schwarz matrix are real. These findings provide insights into the stability properties of systems characterized by Schwarz matrices and offer new tools for the analysis of interconnected subsystems in a cascaded structure.
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